1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CHAPTER 22:
MORPHODYNAMICS OF RECIRCULATING AND FEED FLUMES
Laboratory flumes have proved to be
valuable tools in the study of sediment
transport and morphodynamics. Here the
case of flumes with vertical, inerodible
walls are considered. There are two basic
types of such flumes:
• Recirculating flumes and
• Feed flumes
In addition, there are several variant types,
one of which is discussed in a succeeding
slide.
Recirculating flume in the
Netherlands used by A. Blom and M.
Kleinhans to study vertical
sediment sorting by dunes. Image
courtesy A. Blom.
1
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
THE MOBILE-BED EQUILIBRIUM STATE
The flume considered here is of the simplest type; it has a bed of erodible alluvium,
constant width B and vertical, inerodible walls. The bed sediment is covered by
water from wall to wall. One useful feature of such a flume is that if it is run long
enough, it will eventually approach a mobile-bed equilibrium, as discussed in
Chapter 14. When this state is reached, all quantities such as water discharge Q
(or qw = Q/B), total volume bed material sediment transport rate Qt (or qt = Qt/B),
bed slope S, flow depth H etc. become spatially constant in space (except in
entrance and exit regions) and time. (The parameters in question are averaged
over bedforms such as dunes and bars if they are present.)
Neglect entrance and exit regions
water
H
sediment
S
pump
2
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
THE EQUILIBRIUM STATE: REVIEW OF MATERIAL FROM CHAPTER 14
The hydraulics of the equilibrium state are those of normal flow. Here the case of a
plane bed (no bedforms) is considered as an example. The bed consists of uniform
material with size D. The governing equations are (Chapter 5):
Water conservation:
qw  UH
Momentum conservation:
b  gHS
Friction relations:
1/ 6
b  Cf U2
C f  const (Chezy )
or
Cf 1/ 2
H
  r  
 kc 
(Manning  Strickler )
where kc is a composite bed roughness which may include the effect of bedforms
(if present).
Generic transport relation of the form of Meyer-Peter and Müller for total bed
material load: where t and nt are dimensionless constants:
 

  t  b  c 
RgD D
 RgD

qt
nt
3
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
THE EQUILIBRIUM STATE: REVIEW contd.
In the case of the Chezy resistance relation, for example, the equations governing
the normal state reduce to:

 C f q2w
H  
 gS
1/ 3



 C q2
qt  RgD D  t  f w
 g
1/ 3



S 2 / 3   
  c 
RD 


nt
In the case of the Manning-Stickler resistance relation, the equations
governing the normal state reduce with to:
 k1c/ 3 q2w
H   2
  r gS



3 / 10
1/ 3 2

 k c qw
qt  RgD D  t  2
 g

 r



3 / 10

nt


S

  c 
RD 



7 / 10
Let D, kc and R be given. In either case above, there are two equations for four
parameters at equilibrium; water discharge per unit width qw, volume sediment
discharge per unit width qt, bed slope S and flow depth H. If any two of the set (qw,
qt, S and H) are specified, the other two can be computed.
4
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
THE RECIRCULATING FLUME
In a recirculating flume all the water and all the sediment are recirculated through a
pump. The total amounts of water and sediment in the system are conserved. In
addition to the sediment itself, the operator is free to specify two parameters in
operating the flume: the water discharge per unit width qw and the flow depth H.
The water discharge (and thus the discharge per unit width qw) is set by the
pump setting. (More properly, what are specified are the head-discharge relation of
the pump and the setting of the valve on the return line, but in many recirculating
systems flow discharge itself can be set with relative ease and accuracy.)
water
sediment
The constant flow depth H reached at
equilibrium is set by the total amount of
water in the system, which is conserved.
Increasing the total amount of water in
the system increases the depth reached
at final mobile-bed equilibrium.
pump
Thus in a recirculating flume, equilibrium qw and H are set by the flume operator,
5
and total volume sediment transport rate per unit width qt and bed slope S
evolve to equilibrium accordingly.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
THE FEED FLUME
In a feed flume all the water and all the sediment are fed in at the upstream and
allowed to wash out at the downstream end. Water is introduced (usually pumped)
into the channel at the desired rate, and sediment is fed into the channel using e.g. a
screw-type feeder at the desired rate. In addition to the sediment itself, the operator
is thus free to specify two parameters in the operation of the flume: the water
discharge per unit width qw and the total volume sediment discharge per unit
width qt reached at final equilibrium, which must be equal to the feed rate qtf.
sediment feeder
Thus in a feed flume,
equilibrium qw and qt are
set, by the flume operator,
and equilibrium flow
depth H and bed slope S
evolve accordingly.
water
sediment
pump
tailgate
6
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
A HYBRID TYPE: THE SEDIMENT-RECIRCULATING, WATER-FEED FLUME
In the sediment-recirculating, water-feed flume the sediment and water are separated
at the downstream end. Nearly all the water overflows from a collecting tank. The
sediment settles to the bottom of the collecting tank, and is recirculated with a small
amount of water as a slurry. The water discharge per unit width qw is thus set by
the operator (up to the small fraction of water discharge in the recirculation line). The
total amount of sediment in the flume is conserved. In addition, a downstream weir
controls the downstream elevation of the bed. The combination of these two
conditions constrains the bed slope S at mobile-bed equilibrium. Adding more
sediment to the flume increases the equilibrium bed slope.
recirculated
sediment
water
weir
sedimentfree water
sediment
water
pump
slurry pump
slurry of sediment + small
fraction of water discharge
Thus in a sedimentrecirculating, water-feed
flume, qw and S are set by
the flume operator and qt
and H evolve toward
equilibrium accordingly.
7
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
MORPHODYNAMICS OF APPROACH TO EQUILIBRIUM
The final mobile-bed equilibrium state of a flume is usually not precisely known in
advance. Flow is thus commenced from some arbitrary initial state and allowed to
approach equilibrium. This motivates the following two questions:
•
How long should one wait in order to reach mobile-bed equilibrium?
•
What is the path by which mobile-bed equilibrium is reached?
It might be expected that the answer to these questions depends on the type of flume
under consideration. Here two types of flumes are considered: a) a pure feed
flume and b) a pure recirculating flume.
In performing the analysis, the following simplifying assumptions (which can easily be
relaxed) are made:
1. The flow is always assumed to be subcritical in the sense that Fr < 1.
2. The channel is assumed to have a sufficiently large aspect ration B/H that
sidewall effects can be neglected.
3. Bed resistance is approximated in terms of a constant resistance coefficient
Cf, so that the details of bedform mechanics are neglected.
4. The sediment has uniform size D.
The analysis presented here is based on Parker (2003).
8
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
THE LEGEND OF SEDIMENT LUMPS IN RECIRCULATING FLUMES
pump
In the world of sediment flumes, there is a persistent legend concerning recirculating
flumes that is rarely documented in the literature. That is, these flumes are said to
develop sediment “lumps” that recirculate round and round, either without dissipating
or with only slow dissipation. The author of this e-book has heard this story from T.
Maddock, V. Vanoni and N. Brooks. One of the author’s graduate students
encountered these lumps in a recirculating, meandering flume and showed them to
9
the author (Hills, 1987).
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
THE LEGEND OF SEDIMENT LUMPS IN RECIRCULATING FLUMES
pump
In the world of sediment flumes, there is a persistent legend concerning recirculating
flumes that is rarely documented in the literature. That is, these flumes are said to
develop sediment “lumps” that recirculate round and round, either without dissipating
or with only slow dissipation. The author of this e-book has heard this story from T.
Maddock, V. Vanoni and N. Brooks. One of the author’s graduate students
encountered these lumps in a recirculating, meandering flume and showed them 10
to
the author (Hills, 1987).
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
THE LEGEND OF SEDIMENT LUMPS IN RECIRCULATING FLUMES
pump
In the world of sediment flumes, there is a persistent legend concerning recirculating
flumes that is rarely documented in the literature. That is, these flumes are said to
develop sediment “lumps” that recirculate round and round, either without dissipating
or with only slow dissipation. The author of this e-book has heard this story from T.
Maddock, V. Vanoni and N. Brooks. One of the author’s graduate students
encountered these lumps in a recirculating, meandering flume and showed them 11
to
the author (Hills, 1987).
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
THE LEGEND OF SEDIMENT LUMPS IN RECIRCULATING FLUMES
pump
In the world of sediment flumes, there is a persistent legend concerning recirculating
flumes that is rarely documented in the literature. That is, these flumes are said to
develop sediment “lumps” that recirculate round and round, either without dissipating
or with only slow dissipation. The author of this e-book has heard this story from T.
Maddock, V. Vanoni and N. Brooks. One of the author’s graduate students
encountered these lumps in a recirculating, meandering flume and showed them 12
to
the author (Hills, 1987).
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
THE LEGEND OF SEDIMENT LUMPS IN RECIRCULATING FLUMES
pump
In the world of sediment flumes, there is a persistent legend concerning recirculating
flumes that is rarely documented in the literature. That is, these flumes are said to
develop sediment “lumps” that recirculate round and round, either without dissipating
or with only slow dissipation. The author of this e-book has heard this story from T.
Maddock, V. Vanoni and N. Brooks. One of the author’s graduate students
encountered these lumps in a recirculating, meandering flume and showed them 13
to
the author (Hills, 1987).
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
PARAMETERS
x
t
H
U
=
=
=
=

S
g
qt
=
=
=
=
qw
b
L
B
D
p
=
=
=
=
=
=
streamwise coordinate
time
H(x, t) = flow depth
U(x, t) = depth-averaged flow
velocity
(x, t) = bed elevation
- /x = bed slope
gravitational acceleration
volume bed material sediment
transport rate per unit
width
UH = water discharge per unit
width
boundary shear stress at bed
flume length
H
flume width
sediment size
porosity of bed deposit of sediment
water

sediment
pump
L
sediment feeder
water

sediment
pump
L
tailgate
B
14
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
KEY APPROXIMATIONS AND ASSUMPTIONS
H
• Flume has constant width B.
B
• Sediment is of uniform size D.
• H/B << 1: flume is wide and sidewall
effects can be neglected.
• Flume is sufficiently long so that
entrance and exit regions can
be neglected.
• Flow in the flume is always Froudesubcritical: Fr = U/(gH)1/2 < 1.
• qt/qw << 1: volume transport rate of
sediment is always much lower
than that of water.
• Resistance coefficient Cf is approximated
as constant.
water

sediment
pump
L
sediment feeder
water

sediment
pump
L
tailgate
15
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
GOVERNING EQUATIONS: 1D FLOW
H UH

0
t
x
UH U2H
1 H2
 b

 g
 gH 
t
x
2 x
x 
(1  p )

q
 t
t
x
b  CfU2
Flow mass balance
Flow momentum balance
Sediment mass balance
Closure relation for shear stress: Cf =
dimensionless bed friction coefficient
The condition qt/qw << 1 allows the use of the quasi-steady approximation
introduced in Chapter 13, according to which the time-dependent terms in the
16
equations of flow mass and momentum balance can be neglected.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
REDUCTION TO BACKWATER FORM
The equations of flow mass and momentum balance reduce to the standard
backwater equation introduced in Chapter 5.
UH
0
x
Uh  qw
 UH  qw
,
U2H
1 H2

 g
 gH  CfU2
x
2 x
x

H  
q2w  
q2w 
 1 

  
 Cf
3 
3 
gH   gH 
x  x
or thus
H S  Sf

2
x 1  Fr
where
2
2
q
U
Sf  CfFr 2 , Fr 2  w3 
gH
gH
17
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SEDIMENT TRANSPORT RELATION
Sediment transport is characterized in terms of the same generic sediment transport
relation as used in Chapter 20, except that the parameter s is set equal to unity.
Thus where
D
s

R
=
=
=
=
grain size (uniform)
sediment density
water density
(s/ ) – 1  1.65

0 if

qt  

 nt
t (  c )
qt
q 

RgD D

t
Einstein number
  c


c
 
if
b
 

RgD

Shields number
18
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CONSTRAINTS ON A RECIRCULATING FLUME
Water discharge qw is set by the pump.
The total amount of water in the flume is conserved.
With constant width, constant storage in the return line and
negligible storage in the entrance and exit regions (L sufficiently
large), the constraint is (where C1 is a constant):
L
 Hdx  C
0
1
At final equilibrium, when H = Ho, the constraint reduces to HoL =
C1, according to which Ho is set by the total amount of water.
The total amount of sediment in the flume is conserved.
Neglecting storage in the return line and the head box, the
constraint is (where C2 is another constant):
Integrate the

q
equation of sediment (1   p )
 t
t
x
mass conservation
But from above
d L
dx  0

0
dt
to get
(1  p )
So a cyclic boundary
condition is obtained:

L
0
dx  C2
d L
dx  qt (0)  qt (L)

0
dt
qt (0)  qt (L)
19
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
Three constraints:
© Gary Parker November, 2004
CONSTRAINTS ON A FEED FLUME
Water discharge qw is set by the pump.
The upstream sediment discharge is set by the feeder.
Where qtf is the sediment feed rate:
Let  =  + H denote water surface elevation.
The downstream water surface elevation d is set by the
tailgate:
qt x 0  qtf
  H xL  d
The long-term equilibrium approached in a recirculating flume (without lumps)
should be dynamically equivalent to that obtained in a sediment-feed flume
(e.g. Parker and Wilcock, 1993).
20
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
MOBILE-BED EQUILIBRIUM
The equation for water conservation reduces to:
UH  qw
At mobile-equilibrium the equation of momentum balance reduce to the relation for
normal flow introduced in Chapter 5:
U
H

U2
U
 g
 g  Cf

x
x
x
H
CfU2  gHS

S
x
The sediment transport relation reduces to the form:

C fU2
0 if
 c

RgD
qt


NL
2
CfU2
RgD D   C fU

L 
 c 
if
 c
  RgD
RgD

which applies whether or not mobile-bed equilibrium is reached.
Let R, g, D, t, nt, c* and Cf be specified. The equations in the boxes define
21
three equations in five parameters Uo, Ho, qto, qw and So at mobile-bed equilibrium.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
MOBILE-BED EQUILIBRIUM contd.
Recirculating flume:
Water discharge/width qw is set by pump.
The subscript “o” denotes mobilebed equilibrium conditions.
Total amount of water Vw in system is
conserved. Assuming constant storage in
return line and neglecting entrance and exit
storage, Vw = HLB  depth Ho is set.
water

sediment
pump
Solve three equations for qto, U0, So.
L
Feed flume:
sediment feeder
Water discharge/width qw is set by pump.
water

sediment
Sediment discharge/width qto is set by
feeder.
Solve three equations for uo, Ho, So.
pump
L
tailgate
22
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
APPROACH TO EQUILIBRIUM: NON-DIMENSIONAL FORMULATION
Dimensionless parameters describing the approach to equilibrium (denoted by the
tilde or downward cup) are formed using the values Ho, qto, So corresponding to
normal equilibrium.
~
H  HoH
qt  qto~
q

x  Lx
SoL2 
t  (1  p )
t
qto
Bed elevation  is decomposed into into a spatially averaged value ( t ) and a
deviation from this d(x, t), so that by definition
    d ,

L
0
ddx  0
The above two parameters are made dimensionless as follows:;
 
~
  Hoa (t̂) , d  SoLd (x, t )
From the above relations,
where
Ho
Fl 
SoL

  SoL ,
 
  d  Fl ~
a
= the dimensionless flume number
23
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
APPROACH TO EQUILIBRIUM contd.
The dimensionless relations governing the approach to equilibrium are thus as
follows; where Fro and o denote the Froude and Shields numbers at mobile-bed

equilibrium, respectively,
 ~
1/ 2
~  d  H3
2



H
q
The backwater relation
w
x


Fl

,
Fr


o
~
3 

2
3
is:
x
1  Fro H
 gHo 
d~

Fl a  ~
q x 0  ~
q x 1
dt
The relation for sediment conservation



~
q
decomposes into two parts:
d
    ~
q x 0  ~
q x 1
x
t

The sediment transport relation
is :
where
 
1
~

0 if H2  r

NL
 ~ 2
1
~



q   H  r
~ 2

if
H




r

 1
1



r

 
 
 
o
  
c

r
 
CfUo2
Cf q2w
,  

RgD RgDHo2

o

1
24
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
APPROACH TO EQUILIBRIUM IN A FEED FLUME
In a feed flume, the boundary condition on the sediment transport rate at the
upstream end is qt x 0  qtf  qto , or in dimensionless variables,
~
q x 0  1
Thus the relations for sediment conservation of the previous slide reduce to
d~

Fl a  1  ~
q x 1 ,
dt

d
~
q
     1 ~
q x 1
x
t


The boundary condition on the backwater equation is
  H xL  d
where d is a constant downstream elevation set by a tailgate. This condition
must hold at all flows, including the final mobile-bed equilibrium. Now the datum
for elevation is set (arbitrarily but conveniently) to be equal to the bed elevation
at the center of the flume (x = 0.5 L) at equilibrium, so that ao = 0 and

1
So x  Ho  d
2
Between the above two relations and the nondimensionalizations, it is found that
1 1 
~

25
H   1 ~
a 
  d x 1 
x 1
FL  2

1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
APPROACH TO EQUILIBRIUM IN A FEED FLUME: SUMMARY
The equations

d ~ 3
~   H
H
Fl   x 2 ~ 3
x
1  Fro H
,
d~
a
Fl   1  ~
q x 1 ,
dt

d
~
q
     1 ~
q x 1
x
t


must be solved with the sediment transport relation and boundary conditions:
1
~

0 if H2  r
1 1 
~


NL
~
~
 ~ 2

1
~
H

1







q

1
,

a
d x 1 

q    H  r 
,
~ 2
x 0
x 1
 1
FL  2

if H  r



1

1




r



 
 
 
 
and a suitable initial condition, e.g. where SI is an initial bed slope and aI is an
initial value for flume-averaged bed elevation, , a = aI, d = SIL[0.5 – (x/L)], or in
dimensionless terms.
~
a t 0  ~
aI
SI  1  


, d t 0 
  x
So  2

Note furthermore that aI ( ~
aI ) and SI must be chosen so as to yield subcritical flow
26
in the flume at t = 0 ( t  0 ).
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
APPROACH TO EQUILIBRIUM IN A FEED FLUME: FLOW OF THE CALCULATION
In the case of a feed flume, the calculation flows directly with no iteration. At any time


(e.g. t  0 ) the bed elevation profile, e,g, the parameters ~a and d , are known. The

backwater equation
 ~ 3
~  d  H
H
Fl   x 2 ~ 3
x
1  Fro H
,
~
is then solved for H
using the standard step method of Chapter 5 upstream from the
downstream end, where the boundary condition is:
1 1 
~

H   1 ~
a 
  d x 1 
x 1
FL  2

~
Once H
is known everywhere, ~q is obtained everywhere from the sediment transport
relation of the previous slide. The bed elevation one time step later is determined
from a discretized version of
d~

Fl a  1  ~
q x 1 ,
dt

d
~
q
     1 ~
q x 1
x
t


where the second equation above is solved subject to the boundary condition
~
q x 0  1
27
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
APPROACH TO EQUILIBRIUM IN A RECIRCULATING FLUME
As shown at the bottom of Slide 19, the sediment transport in a recirculating flume
must satisfy a cyclic boundary condition. In dimensionless terms, this becomes
~
q x 1  ~
q x 0
As a result, the relations for sediment conservation of Slide 24 reduce to
d~

Fl a  ~
q x 0  ~
q x 1  0 ,
dt

d
~
q ~
~
q
~
     q q   
x 0
x 1
x
x
t


The total amount of water in the flume is conserved. Evaluating the constant in
the equation at the top right of Slide 19 from the final equilibrium, then,
L
 Hdx  H L
o
0
In dimensionless form, this constraint becomes

1
0
~ 
Hdx  1
28
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
APPROACH TO EQUILIBRIUM IN A RECIRCULATING FLUME: SUMMARY
The equation for ~
a can be dropped because flume-averaged bed elevation cannot
change in a recirculating flume. The remaining backwater and sediment
conservation relations

d ~ 3
~   H
H
Fl   x 2 ~ 3
x
1  Fro H
,

d
~
q
  
x
t
must be solved with the sediment transport relation and constraints:
1
~

0 if H2  r
1~ 
~
~

Hdx  1
NL
 ~ 2
q

q
,
1
~



0
q    H  r 
,
x 0
x 1
~ 2
 1
if H  r


 1


  1  r
 
 
 
The initial condition is
 

SI  1  


d t 0 
  x
So  2

where the initial slope SI must be chosen so as to yield subcritical flow everywhere.
29
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
APPROACH TO EQUILIBRIUM IN A RECIRCULATING FLUME: FLOW OF THE
CALCULATION
The method of solution is similar to that for the feed flume with one crucial difference:
iteration is required to solve the backwater equation over a known bed

d ~ 3
~   H
H
Fl   x 2 ~ 3
x
1  Fro H
subject to the integral condition

1
0
~ 
Hdx  1
That is, for any guess
~
~
Hd  H 
x 1
it is possible to solve the backwater equation and test to see if the integral
~
condition is satisfied. The value of Hd necessary to satisfy the backwater equation
can be found by trial and error, or as shown below, by a more systematic set of
30
methods.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
RESPONSE OF A RECIRCULATING FLUME TO AN INITIAL BED SLOPE
THAT IS BELOW THE EQUILIBRIUM VALUE
The low initial bed slope causes the depth to be too high upstream. Total water mass in
the flume can be conserved only by constructing an M2 water surface profile. The result
is a shear stress, and thus sediment transport rate at the downstream end that is higher
than the upstream end. This sediment is immediately recirculated upstream, where it
cannot be carried, resulting in bed aggration there.
initial water surface profile (M2)

ultimate bed profile
initial bed profile

initial profile of sediment transport
q
31
x
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
RECIRCULATING FLUME: ITERATIVE SOLUTION OF BACKWATER EQUATION
The shooting method is combined with the Newton-Raphson method to devise a
~
scheme to solve the backwater equation iteratively. Now for each guess Hd it is
possible to solve the backwater equation

d ~ 3
~   H
H
Fl   x 2 ~ 3
x
1  Fro H
~
for H , so that in general the solution can be written as
~ ~~ 
H  H(Hd, x )
~
Now define the function (H
such that
d)
1~ ~ 

~
(Hd )   H(Hd, x ) dx  1
0
~
The correct value of H
is the one for which the integral constraint is satisfied, i.e.
d
1~ ~ 

~
(Hd )   H(Hd, x ) dx  1  0
0
32
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
ITERATIVE SOLUTION OF BACKWATER EQUATION contd.
~( w ) is an estimate of the solution
Now according to the Newton Raphson method, if H
d
~ ( w 1)
~
for Hd , then a better estimate Hd (where w = 1, 2, 3… is an iteration index) is given
as
(H(dw ) )
~( w 1)
(w)
Hd
 Hd 
d ( w )
(Hd )
dHd
Reducing this relation with the definition of ,
1~ ~ 

~
(Hd )   H(Hd, x ) dx  1
0
results in the iteration scheme
~ ~( w )  
H(Hd , x )dx  1

( w)
0
 Hd 
~
1 H ~

(w) 
(
H
,
x
)
d
x
0 H~ d
d
1
~
H(dw 1)
~
~
This iterative scheme requires knowledge of the parameter H / Hd .
33
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
ITERATIVE SOLUTION OF BACKWATER EQUATION contd.
Now define the variational parameter Hv as
~
H
~ 
Hv (Hd, x)  ~
Hd
The governing equation and boundary condition of the iterative scheme are

d ~ 3
~   H
H
Fl   x 2 ~ 3
x
1  Fro H
~
~
, H   Hd
x 1
~
Taking the derivative of both equations with respect to Hd results in the variational
equation and boundary condition below;

d ~ 3
~
 H
Hv
3H 4
2 x
Fl  
[
1

Fr
]Hv
o
2 ~ 3
2 ~ 3
x
1  Fro H
1  Fro H
, Hv (1)  1
34
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
ITERATIVE SOLUTION OF BACKWATER EQUATION contd.
~
For any guess H(dw ) , then, the following two equations
and boundary conditions can

~( w )
be used to find H
and H(vw ) for all values of x .

d ~ ( w ) 3
~
   (H )
H( w )
~( w )
~( w )

x
Fl  
, H   Hd
2 ~ ( w ) 3
x 1
x
1  Fro (H )

d ~ ( w ) 3
~
  (H )
H(vw )
3(H( w ) ) 4
2 x
(w)
(w)
Fl  
[
1

Fr
]
H
H
o
v
v (1)  1
2 ~ ( w ) 3
2 ~ w ) 3
x
1  Fro (H )
1  Fro (H )
~
The improved guess H(dw 1) is then given as
~( w ) 
H
dx  1

( w)
0
 Hd  1
(w) 
H
 v dx
1
~
H(dw 1)
0
The iteration scheme is continued until convergence is obtained.
35
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
NUMERICAL IMPLEMENTATION
Numerical implementations of the previous formulation for both recirculating and feed
flumes are given in RTe-bookRecircFeed.xls. The GUI for the case of recirculation is
given in worksheet “Recirc”, and the GUI for the case of feed is given in worksheet
“Feed”. The corresponding codes are in Module 1 and Module 2.
Both these formulations use a) a predictor-corrector method to compute backwater
curves, and b) pure upwinding to compute spatial derivatives of qt in the various
Exner equations of sediment conservation. The discretization given below is identical
to that used in Chapter 20.
 1
x 
M
ghost
i=1
2
3


xi  (i  1)x
M -1
, i  1..M  1
M
i = M+1
M+1
1
The ghost node for sediment feed is not used, however, in the implementation
for the recirculating flume.
36
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
NUMERICAL IMPLEMENTATION contd.
The backwater equation is solved using a predictor-corrector scheme. The solution
proceeds upstream from the downstream node M+1, where for a feed flume
1 1 
~

HM1  1  ~
a 
  d,M1 
FL  2

~
and for a recirculating flume HM1 is obtained from the iteration scheme using the
Newton-Raphson and shooting techniques. The discretized backwater forms for
~
dimensionless depth H
are:
~
Hpred

d,i1  ˆ d,i ~ 3
 Hi


~ 1

x
 Hi 

x
~
Fl 1  Fro2 Hi3




d,i1  i ~ 3 d,i1  d,i ~ 3
 Hi
 Hpred 


1
~
~

x

x
, Hi1  Hi 
[
]x
~ 3 
2 ~ 3
2 Fl 1  Fro2 H
1  Fro Hpred
i
In the case of a recirculating flume, the variational parameter Hv must also be
computed subject to the boundary condition Hv,M+1 = 1. The corresponding forms are
Hv,pred
ˆ d,i1  ˆ d,i ~ 3
~ 4

(
 Hi ) 

1 3Hi Hv,i
2

x
 Hv,i 
]x
~ 3 [1  Fro
2 ~ 3
Fl 1  Fro2 H
1

Fr
H
i
o
i
ˆ d,i1  ˆ d,i ~ 3
ˆ d,i1  ˆ d,i ~ 3
~
~ 4
4

(

H
)

(
 Hi1) 


i
3Hi1Hv,pred
3Hi Hv,i
1
2
2

x

x
Hv,i1  Hv,i 
{
]
[1  Fro
]} x
~ 3 [1  Fro
2 ~ 3
2 ~ 3
2 ~ 3
37
2 Fl 1  Fro2 H
1

Fr
H
1

Fr
H
1

Fr
H
i
o
i
o
i1
o
i1
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
NUMERICAL IMPLEMENTATION contd.
In the case of a feed flume, the discretized forms for sediment conservation are

1
~
~
~
a t  t  a t  1  qM1  t
Fl



t

~
~
d,i t  (1  qi )   1  qM1  t , i  1


x
d,i t  t  


t

~
~
d,i   ( qi1  qi )   1  q̂M1  t , i  2..M  1
 t
x
In the case of a recirculating flume, the corresponding form is

d,i t  t

 
t
~
~



(
q

q
)
 , i1
d,i t
M1
i

 x

t

~
~


 d,i t  ( qi1  qi ) x , i  2..M  1
38
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CRITERION FOR EQUILIBRIUM
It is assumed that equilibrium has been reached when the bed slope is everywhere
within a given tolerance of the equilibrium slope So. A normalized bed slope SN that
everywhere equals unity at equilibrium is defined as

S
1  d
SN 

 
So
So x
x
For the purpose of testing for convergence, the normalized bed slope SN,i at the ith
node is defined as


i1  i
SN,i 
, i  2..M  1

x
At equilibrium, then, SN,i should everywhere be equal to unity. The error i between
the bed slope and the equilibrium bed slope at the ith node is
i 
Convergence is realized when
SN,i  1
(SN,i  1) / 2
max( i )  t
where t is a tolerance. In RTe-bookRecircFeed.xls t has been set equal to
0.01 (parameter “epslope” in a Const statement).
39
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
PHASE PLANE INTERPRETATION
One way to interpret the results of the analysis is in terms of a phase plane. Let Sup
denote the bed slope at the upstream end of the flume, and Sdown denote it at the
downstream end, where






d,1  d,2
d,M  d,M1
d
d
Sup  So 
 So
, Sdown  So 
 So


x x 0
x
x x 1
x
Further define the normalized slope SN as equal to S/So, so that
SN,up
Sup

So
, SN,down 
Sdown
So
The initial normalized slope is denoted as SNI. At mobile-bed equilibrium slope S is
everywhere equal to the equilibrium value So, so that SN = 1 everywhere. That is,
one indicator of mobile-bed equilibrium is the equality
(SN,up ,SN,down )  (1,1)
In a phase plane interpretation, SN,up is plotted against SN, down at every time step, and
the approach toward (1, 1) is visualized. This equilibrium point (1, 1) is called the
fixed point of the phase problem.
40
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
PHASE PLANE INTERPRETATION contd.
The initial condition for the bed profile is S = SI (SN = SNI) everywhere, where SI is not
necessarily equal to the mobile-bed equilibrium value So (SNI is not necessarily equal
to 1) . For example, in the case SI = 0.5 So, (SN,up, SNdown) begin with the values (0.5,
0.5) and gradually approach (1, 1).
An approach in the form of a spiral is usually indicative of damped sediment waves,
or lumps.
2
fixed point
(equilibrium)
SN,up 1
initial point
0
0
1
SN,down
2
41
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
INPUT PARAMETERS USED IN THE CODES
Input for recirculating flume
Fl 
Ho
SoL
SNI 
Input for feed flume
SI
So
SI = initial bed slope
So = equilibrium bed slope
42
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SAMPLE CALCULATION WITH RTe-bookRecircFeed.xls: RECIRCULATING FLUME
Fro
nt
Fl
taur
SNI
M
dt
Nstep
Ntimes
0.4
1.5
10
3
0.5
50
0.0025
300
10
This calculation requires
 a
dimensionless time t of
11.94 in order to reach
mobile-bed equilibrium. The
spiralling is indicative of a
damped sediment wave or
lump. This is shown in more
detail in the next slide.
2
final equilibrium
1.5
SNup
Fro
nt
Fl
o*/c*
SNI
M
t
1
0.5
initial
0
0
0.5
1
SNdown
1.5
2
43
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
BED EVOLUTION IN RECIRCULATING FLUME
1
 



Profiles
from
to
t

0
t  to
0.75
Profiles from t  0.75
t  1.50
Dimensionless elevation
0.8
 t  2.25 to t 3.00
Profiles from
Profiles
from
t  1t.50
 2.t25
Profiles
from
 3to
.00t to
 3.75
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
0.2
0.4
0.6
0.8
1
Dimensionless distance
44
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SAMPLE CALCULATION WITH RTe-bookRecircFeed.xls: FEED FLUME
~
aI
o*/c*
SNI
M
t
Fro
nt
Fl
etaatil
taur
SNI
M
dt
Nstep
Ntimes
0.4
1.5
10
0
3
0.5
50
0.0025
100
10
The input parameters are
comparable to that of the
previous case of recirculation.
Mobile-bed equilibrium is
reached
 in a dimensionless
time t of only 4.33. The
phase diagram shows no
spiralling.
2
final equilibrium
1.5
SNup
Fro
nt
Fl
1
0.5
initial
0
0
0.5
1
1.5
2
SNdown
45
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
BED EVOLUTION IN FEED FLUME
Dimensionless elevation
0.6




 1.50
Profiles
from
to
t

0
.
75
t

Profiles
from
to to
t t0.275
Profiles
fromtt01.50
.25
Profiles from t  2.25 to t  3.00
0.4
0.4
0.2
0.2
0
0
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6
-0.8
0
0.2
0.4
0.6
0.8
1
Dimensionless distance
46
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
APPLICATION EXAMPLE
The dimensionless numbers of the previous two calculations, i.e. Fro = 0.4, nt = 1.5,


aI = 0 (feed case only) are now converted to
Fl = 10, 0 / c = 3, SNI = 0.5 and ~
dimensioned numbers for a sample case, for which
D = 1 mm, R = 1.65
ks = 2.5 D
Ho = 0.2 m (equilibrium depth)
The sediment is assumed to move exclusively as bedload. The assumed bedload
relation, given below, is from Chapter 6.
qt  qb  3.97 (  c )3 / 2
, c  0.0495
The assumed resistance relation, given below, is from Chapter 5.
Cz  Cf 1/ 2  8.1H ks 
1/ 6
From Ho, ks and the above resistance relation,
From Ho and Fro = 0.4,


From 0 / c = 3,
From o*, D, R and the load relation,
From o* = HoSo/(RD), Ho, R and D,
From Fl = Ho/(SoL), Ho and So,
Cf = 0.00354
qw = 0.560 m2/s
o* = 0.148
qto = 1.57 x 10-5 m2/s
So = 0.00122
L = 16.3 m
47
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
APPLICATION EXAMPLE contd.
In the example, then, the equilibrium parameters are
qw = 0.560 m2/s
Ho = 0.2 m
So = 0.00122
qto = 1.57 x 10-5 m2/s (= 1250 grams/minute for a flume width B = 0.5 m
and the flume length is
L = 16.3 m
The
 time to equilibrium tequil is related to the corresponding dimensionless parameter
tequil as
2 
t equil  (1  p )
SoL
tequil
qto
Assuming the value for bed porosity p of 0.4, the time to equilibrium for the
rercirculating case of Slide 43 and the feed case of Slide 45 are then
Recirculating flume:
Feed flume:
tequil = 41.3 hours
tequil = 15.0 hours
48
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SOME CONCLUSIONS
The following tentative conclusions can be reached concerning
sediment lumps in flumes.
1. Cyclic lumps do occur in recirculating flumes.
2. These lumps do eventually dissipate.
3. Similar cyclic lumps are not manifested in sediment-feed flumes.
4. Feed flumes reach mobile-bed equilibrium faster than recirculating
flumes.
5. Both flume types eventually reach the same mobile-bed equilibrium.
49
1D SEDIMENT
TRANSPORT
MORPHODYNAMICS
1D SEDIMENT
TRANSPORT
MORPHODYNAMICS
with applications to
with applications to
RIVERS AND TURBIDITY CURRENTS
RIVERS
AND
TURBIDITY
CURRENTS
© Gary Parker
November,
2004
REFERENCES FOR CHAPTER 22
Hills, R., 1987, Sediment sorting in meandering rivers, M.S. thesis, University of Minnesota, 73 p.
+ figures.
Parker, G., 2003, Persistence of sediment lumps in approach to equilibrium in sedimentrecirculating flumes, Proceedings, XXX Congress, International Association of Hydraulic
Research, Thessaloniki, Greece, August 24-29, downloadable at
http://cee.uiuc.edu/people/parkerg/conference_reprints.htm .
Parker, G. and Wilcock, P., 1993, Sediment feed and recirculating flumes: a fundamental
difference, Journal of Hydraulic Engineering, 119(11), 1192-1204.
50